The Alexander Polynomial and Knot Floer Homology Robert Lipshitz1 University of Oregon

Home , Conway knot, Ribbon knot, Seifert surface

The Alexander polynomial and knot Floer homology Robert Lipshitz1 University of Oregon

(Mostly work of other people. Partly joint with Peter Ozsváth and Dylan Thurston, or David Treumann, or Kristen Hendricks and Sucharit Sarkar.)

1. RL was supported by NSF grant DMS-1149800. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation." Seifert surfaces • Every knot � in � bounds an orientablesurface. • Definition. Minimal genus of such a surface is �(�). • Definition. An orientable surface with boundary � is a Seifert surface for �. Seifert matrix and Alexander polynomial • � -- a Seifert surface for �. • Let {�} -- a basis for �(�). Seifert matrix and Alexander polynomial • � -- a Seifert surface for �. • {� } -- a basis for �(�). • � be a positive pushoff of � .

• Definition. Seifert matrix � = (�,) with �, = �� , � . −� −� • Example: 0 −2 Seifert matrix and Alexander polynomial • � -- a Seifert surface for �. • {� } -- a basis for �(�). • � be a positive pushoff of � .

• Definition. Seifert matrix � = (�,) with �, = �� , � . −2 −1 • Example. � −� Seifert matrix and Alexander polynomial • � -- a Seifert surface for �. • {� } -- a basis for �(�). • � be a positive pushoff of � .

• Definition. Seifert matrix � = (�,) with �, = �� , � . −2 −1 • Example: 0 −2 • Definition. Alexander polynomial Δ � = ±� det �� − � . • Example: −2� + 2 −� Δ � = ±� 1 −2� + 2 = ±� −4� + 7� − 4 = 4� − 7 + 4�.

• Corollary. 2� � ≥ ��ℎ(Δ � ) Knot Floer homology (Ozsváth-Szabó, Rasmussen, 2003) • � ⊂ � a knot → bigraded abelian group • Also works for null- ��,(�). homologous knots in other 3- manifolds. • ∑ −1 �� = Δ (�) , , • There are also versions for • Homology of a chain complex links. (��, � ,�: ��, � → ��, � ). • There are other variants -- ��(�) ��(�) • Recall: ��ℎ Δ � /2 ≤ �(�). , , etc. • Theorem. (Ozsváth-Szabó) • There are extensions to sutured manifolds (Juhász, , ≔ max � �� ≠ 0} = � � . ∗, Alishahi-Eftekhary). -2 -1 0 1 2 3 � � -2 -1 0 2 � � 1 � 1 Δ � = 1 0 � � 0 � Δ � = −� + 1 − � � � -1 � -1 -2 � � Kinoshita-Terasaka knot Definition of knot Floer homology

• Via pseudo-holomorphic curves (solutions to particular nonlinear PDE’s) in a high-dimensional auxiliary space. • There are now combinatorial definitions (Manolescu-Ozsváth-Sarkar 2006, Manolescu-Ozsváth-Szabó-Thurston 2006, …) • No classical definition is known. (In particular, not the singular homology of any naturally associated space.) Fibered knots • A knot is fibered if it has an � family of Seifert surfaces. • Lemma. If � is fibered, fiber �, monodromy � then Δ(�) is the characteristic polynomial of �∗:� � → �(�).

• Corollary. If � is fibered then Δ(�) is monic and ��ℎ Δ � = 2�(�). • Theorem. (Ozsváth-Szabó, Ghiggini, Ni) �� (�) is monic if and only if � is fibered. • (Monic means ⊕ ��, � ≅ �.)

• Bordered Floer homology (L-Ozsváth-Thurston): • Surface F ⟿ dg algebra A(F). � � � (� Σ ) 164 diml. • � � = �. � � = /�� = �� = 0 ∗ � Lifting the characteristic polynomial formula

�

� Lifting the characteristic polynomial formula

• Bordered Floer homology (L-Ozsváth-Thurston): • Surface F ⟿ dg algebra A(F). • � � = �. � � 10-dimensional, �∗(� Σ ) 164 dimensional. • Cobordism �, � :(�, ��) → (�, ��) ⟿ dg bimodule�� (�, �). • Composition ⟿ derived tensor product.

�

� Lifting the characteristic polynomial formula

�

� Lifting the characteristic polynomial formula

• Recall: for K fibered, Δ(�) is the characteristic polynomial of � : � � → � (�). ∗ ��∗ �� (�, �) �� (� , �) • � =mapping cylinder of � then � � L-Ozsváth-Thurston, Petkova, ∗ ∗ Trace �∗: Λ � � → Λ �(�) Δ(�) Hom-Lidman-Watson: Symmetries Periodic and Freely Periodic Knots

• K is n-periodic if it is preserved by rotation by 2�/� around an axis. • K is freely periodic of period (p,q) if it is preserved by �, � ↦ (� �, � �) on � = { �, � ∈ �| � + � = 1}

T(4,7) is 7- periodic, and freely (4,7) 74 is 2-periodic with periodic. quotient the unknot U. Edmonds’s and Murasugi’s Conditions

• Theorem. (Edmonds, 1984) If K is n-periodic then there is a minimal genus Seifert surface for K preserved by Z/nZ. • (Proof uses minimal surfaces.) • Corollary. If � is the quotient knot then � � ≥ �� + (� − 1)(� − 1)/2 (where � = linking number with axis). Edmonds’s and Murasugi’s Conditions

• Theorem. [(Hendricks, 2012) + �(Hendricks-L-Sarkar 2015)] If K is 2- periodic then there is a spectral sequence �� ∪ � ⊗ �� ⊕ � ⊗ �[�, � ] ⇒ �� ∪ � ⊗ ��[�, � ]. • Proof uses Seidel-I. Smith’s quantum (P.A.) Smith Theory. • Implies (lifts) 2-periodic cases of Edmonds’s Corollary and Murasugi’s Theorem. • Has other implications as well. Branched double covers

• The double cover of � branched along � is �: Σ �,� , � → (�, �) • �| , ∖ : Σ � ,� ∖ � → � ∖ � is a 2-fold cover. • In the normal planes to K, � is � ↦ �. • Special cases: • Σ �, � = (�, �). • If (�, �) is 2-periodic then � ∖ �,� = Σ � ∖ �,� . • Smith Conjecture (proved in 1980’s) states that if � ≠ � then Σ �, � ≠ �. • Can also talk about Σ(�, �), but not always unique. Knot Floer homology in branched double covers • Classical?: For Σ � , � → (� , �), Δ � ≡ Δ(�) (�� 2). • Theorem. (Hendricks, 2011) There is a spectral sequence �� Σ � ,� , � ⊗ � �, � ⇒ �� , � ⊗ � �, � .

• Theorem. (L-Treumann, 2012) If � � = 0 and � ⊂ � has � � ≤ 2 then there is a spectral sequence �� Σ �, � , � ⊗ � �, � ⇒ �� , � ⊗ � �, � . Proof sketch via bordered Floer.

• Cut Y along Seifert surface F for K to get cobordism Z from F to F. • (� ,�) is self-gluing of Z. (Σ � ,�) is self-gluing of � ∪ �. • So, want: �� Σ � , � = ��∗ �� ⊗() ��(�) ⇒ ��∗ �� = ��(� ,�) • (Suppressed copies of �[�, � ].) • Such a spectral sequence exists whenever �(�) satisfies certain algebraic properties. Honest covers • If � → � is a � cover then there is an induced �/2� cover (�/2�) = � → � • Theorem. (L-Treumann, 2012) If � → � is a �/2� cover induced by a � cover then �� ⊗ � ⊕ � ⊗ �[�, � ] ⇒ �� ⊗ �[�, � ]. • Theorem. (Lidman-Manolescu, 2016) If � → � is a �/2� cover and � � = 0 then for each s ∈ ��(�), ∗ �� , � � ⊗ �[�, � ] ⇒ �� , � ⊗ �[�, � ]. • Corollary. �� , �∗� ≥ �� , � . • Lidman-Manolescu is uses a variant of Seiberg-Witten Floer homology. Main work is identifying this variant with more common (and computable) ones. • Some applications of Lidman-Manolescu’s result later in the talk. Symmetric Unions

� �#�(�) �(�) � � • Theorem. (Kinoshita-Terasaka, 1957) If � is even then Δ � = (Δ � ) . • Theorem. (Allison Moore, 2015) If replacing � with � gives a 2-component unlink then �� ≅ �� ⊗ �� (� � ).

• Corollary. � � � = 2�(�) and � � is fibered iff � is. • Corollary. New examples satisfying cosmetic crossing conjecture: any non- nugatory crossing change changes isotopy class of �. More Alexander polynomial formulas • Symmetric knots (K-T): Δ � = (Δ � ) . • If Σ(�) is n-fold cyclic branched cover of � then / � Σ � = ∏ Δ(� ) . e.g., � Σ � = Δ 1 Δ(−1). • If Σ � , � n-fold cyclic branched cover of (�, �) then Δ � = ∏ Δ � � . e.g., if � = 2, Δ � = Δ � Δ(−t). • (Murasugi) If � is n-periodic, axis �, quotient (�, �), � = ��(�, �) then 1 − � Δ � = Δ �, � . 1 − � ∪ • (Hartley, 1981) Similar result for freely periodic knots. • Question. Can any of these be lifted to Floer homology? How? More open questions about Floer homology, Seifert surfaces, and symmetries • There are nice combinatorial definitions of �� (�, �). Can one give combinatorial proofs of key properties, like detecting �(�) or fiberedness? • Edmonds’s condition says a minimal genus Seifert surface is �/�� equivariant. Can one prove this with Floer homology? (Recall that Hendricks proved the main numerical corollary.) • For any knot � in a homology sphere �, is there a spectral sequence �� Σ � ⇒ �� (�)? (cf. Smith conjecture.) Concordance The concordance group and slice genus

• Definition. The smooth slice genus of K is � � = min � � � ⊂ � , �� = � ⊂ � = �� , � ��ℎ} .

� � �

� The concordance group and slice genus

• Definition. The smooth slice genus of K is � � = min � � � ⊂ � , �� = � ⊂ � = �� , � ��ℎ} . • K is slice if � � = 0. • Definition. The smooth concordance group is C= ( �� / �� , #). • Replace “smooth” by “topologically locally flat” gives topological slice genus � (�), topologically slice knots, topological concordance group CTop. • � � ≤ � � , so smoothly slice implies topologically slice.

• 0 →CTS→ C → CTop → 0.

David Eppstein “Square Ribbon Knot” Wikipedia Two classical restrictions

• Recall: the Seifert matrix � = (�, ) with �, = ��(�, � ). • Definition. The signature � � ∈ � is the signature of � + � . • Theorem. (Kauffman-Taylor, Murasugi, Tristram, 1960’s) The signature is a homomorphism CTop → �, and 2� � ≥ �(�). • Theorem. (Fox-Milnor, 1966) If � is topologically slice then there is a polynomial �(�) so that Δ � = � � �(� ). • Theorem. (Levine, 1969) There is a surjection CTop → � ⊕ �/2� ⊕ �/4� . • Question. Is there any torsion other than 2-torsion in C or CTop? Some modern concordance homomorphisms

• Turning to smooth concordance: • Theorem. (Endo, 1995) There is a � subgroup of CTS = ker (C → CTop). • Does it split? Surjections onto �? Bounds on �? • Ozsváth-Szabó, 2003. �: C→ �, via Heegaard Floer homology. Bounds �. • Rasmussen, 2004. s: C→ �, via Khovanov homology. Bounds �. • Manolescu-Owens, 2005. �: C→ �, via Heegaard Floer homology. • �, �, � → � Livingston, 2006. give a surjection CTS . • → � Hom, 2013. A surjection CTS . (Not constructive.) • Υ: → � Ozsváth-Stipsicz-Szabó, 2014. Another surjection CTS . (Constructive.) • Hendricks-L-Sarkar, 2015. Another homomorphism �: C→ �, via equivariantFloer homology. • Many other beautiful results by many other researchers. An explicit example in CTS • Recall: 0 →CTS→ C → CTop → 0. • Theorem. (Freedman-Quinn ~1989) If Δ(�) = 1 then � is topologically slice. • Exercise. The (positive) untwisted Whitehead � (�) � Δ � = 1 double of any knot has ( ) . • Theorem. (Hedden 2006) If � � > 0 then � � � = 1. • Corollary. CTS is nontrivial. • (This is not the first proof, which is attributed to

Casson (unpublished). Gompf gave first examples � � � 2,3 = 1. of non-slice Whitehead doubles.) 4 From CTS to exotic R ’s

• Theorem. (Classical, Moise-Munkres ~1960, Stallings 1961) There is a unique smooth structure on � is � ≤ 2, � = 3, � > 4. • Theorem. (Taubes, 1987) There are uncountably many smooth structures on �. • Theorem. (Freedman-Quinn) Any non-compact 4-manifold has a smooth structure. Here’s an exotic � (Gompf-Stipsicz, Exercise 9.4.23): • Take a knot K in . Let � be result of attaching a 2-handle to � along �. CTS There is a topologically flat embedding � ↪ � . • � ∖ � � Freedman-Quinn implies can be smoothed. Glue to this smoothing of � ∖ � to get a smooth � . Must be exotic, because � is smoothly slice in it. Lifting the Fox-Milnor condition

• Recall: the Fox-Milnor condition: if K is topologically slice then K11n42, mirror of Δ � = � � � � . KT, from Knot Atlas • Question. Is there a lift of this result to �� ? • Presumably such a result would be about smoothly slice knots. • The Kinoshita-Terasaka knot is topologically slice (since Δ � = 1) but �� has Poincaré polynomial � + � � + 4 � + � � + 7 + 6� + 4 � + � � + � + � �, and total rank 2 + 8 + 7 + 6 + 8 + 2 = 33 (not square!). • Maybe some sort of spectral sequence? • Maybe only four doubly slice knots (knots with an invertible concordance to U)? Some more open concordance questions

• Is there a surjection CTS→ (�/2�) ? Any other torsion? • Conjecture. (Kirby 1.38) �(�) is smoothly slice if and only if � is. • Sub-conjecture. The positive Whitehead double of the negative trefoil is not smoothly slice. • What is the smooth slice genus of the Conway knot? (Its mutant, the Kinoshita-Terasaka knot, is slice. Δ � = Δ � = 1.) (From Knot Atlas)

Conway knot Kinoshita-Terasaka knot From Interstellar movie poster (Paramount Pictures)

Dehn surgery is Ouroboros holes.

Adapted from (German) Wikipedia entry, contributed by AnonMoos. Definition of Dehn surgery

• Definition. Dehn surgery on a link � = � ∪ ⋯ ∪ � is � ∖ �� ∪ ⋯ � ∪ � ×� ∪ ⋯ ∪ � ×� . • � � ∖ �� ∪ ⋯ � = � ∐⋯ ∐ � . Surgery is determined by images of the � circles ��×{��} which bound disks �. • Each circle is � ⋅ �� + � ⋅ (��), so have � � . ,⋯, • Theorem. (Lickorish, 1962) Every closed, orientable 3-manifold is � (�) for some , … , and �. ,⋯, Some questions and some answers

• Question. Which manifolds arise as surgery on a knot? • There are some obvious restrictions on �, �. • There are non-obvious restrictions on manifolds with nontrivial first homology (Boyer-Lines 1990). • Theorem. (Hom-Karakurt-Lidman, 2014) There are non-obvious restrictions on manifolds with � = 0. e.g., Σ �, 2� − 1,2� + 1 = �, �, � ∈ � ⊂ � � + � + � = 0 , � ≥ 8 does not. • Question. Are there non-obvious restrictions on which manifolds arise as surgery on an n-component link (n>1 fixed)? More questions and answers

• Question. If � � ≅ � (� ) must � = �′? • Theorem. (Gabai, Kronheimer-Mrowka-Ozsváth-Szabó, 2003) Yes if K=U. • Theorem. (Ghiggini, Ozsváth-Szabó, 2006) Yes if � = 3, �(3), or 4.

Knot Atlas / KnotPlot Surgery characterizations of knots

• Question. If � � ≅ � (� ) must � = �′? • Theorem. (Gabai, Kronheimer-Mrowka-Ozsváth-Szabó, 2003) Yes if K=U. • Theorem. (Ghiggini, Ozsváth-Szabó, 2006) Yes if � = 3, �(3), or 4. • “No” in general. • Question. What about other specific knots (e.g., �,)? Lens space surgeries

• Question. For which � is there �/� with �/(�) a lens space? • Berge Conjecture. Such K is doubly primitive, i.e., • Lies on a genus 2 Heegaard surface • Meets some meridian disk on each side once. • Theorem. (Ni,2006): If �/(�) is a lens space then � is fibered. • Theorem. (Greene, 2010): The lens spaces that arise from knot surgery are those predicted by Berge conjecture. © Ken Baker sketchesoftopology.wordpress.com An application of equivariant Floer homology

• Question. When does one surgery on a knot cover another? • Example. � (3) = �(6� + 1,4�) (Moser, 1971). So � (3) is a (6� + 1)-fold cover of � (3) for � = � + � 6� + 1 �. • Theorem. (Lidman-Manolescu, 2016) If < 1 and �/� < �′/�′ then � (�) is not a regular � sheeted cover of �(�) (� prime). • (Proof uses equivariant Floer homology spectral sequence �� ⇒ �� (�) mentioned earlier.) L-spaces and left orderability L-spaces • Definition. � is a rational homology sphere if �∗ �; � ≅ �∗(� , �). • Theorem. (Ozsváth-Szabó) If � is a rational homology sphere then � �� = � � = �� . • Definition. A rational homology sphere � is an L-space if �� ≅ � (i.e., as small as possible).

• Examples. Space � �� Poincaré homology sphere � (3) 0 � � Lens space � �, � = � (�) �/�� (Any manifold with spherical geometry.) () Σ(�) for � alternating � = det � � = det(� + �) (From Wikipedia) � (� −2,3,7 ) �/�� /�� (Hyperbolic for p>19) Some results for L-spaces • Strangely, Floer theory is very useful for studying L-spaces. Some examples: • Theorem. (Ozsváth-Szabó 2003) If � (�) is an L-space then Δ � = −1 + ∑ −1 (� + � ) for an increasing sequence 0 < � < ⋯ < �. • Strong restriction on what knots have lens space surgeries. • Theorem. (Lidman-Moore 2015) If Σ(�) is an L-space and det (�) is square- free then � satisfies the cosmetic crossing conjecture. • Theorem. (Greene 2011) If �, �′ are alternating and Σ � ≅ Σ(�) then � and �′ are related by a sequence of mutations. • Partial converse to an old observation of Viro. • Uses (grading on) �� Σ � . The L-space = left orderability conjecture

• Definition. A group G is left orderable if there is a total order < on G so that for all f, �, ℎ ∈ �, � < ℎ ⇒ �� < �ℎ. • Example. � is left orderable. • Example. Any finite group is not left orderable. • By convention, the trivial group is not left orderable. © Ken Baker • Conjecture. (Ozsváth-Szabó, Boyer-Gordon-Watson) For irreducible rational homology spheres �, TFAE: Not taut: • � is an L-space. • �(�) is not left orderable. • � does not admit a co-orientable taut foliation.

Reeb foliation Irina Gelbukh / Wikipedia Evidence for the conjecture

• Theorem. (Ozsváth-Szabó 2003) If � admits a co-orientable taut foliation then Y is not an L-space. • Theorem. (Levine-Lewallen 2011) Strong L-spaces are not left orderable. • Many computations. e.g., Boyer-Clay 2015, Hanselman-Rasmussen- Rasmussen-Watson 2015: true for all graph manifolds. • Obvious: if � → � is a finite cover and �(�) is not left orderable then �(�) is not left orderable. • Theorem. (Lidman-Manolescu, 2016) If � → � is a regular, solvable cover, � is an L-space, and �� (�′) is torsion free for all � → � → � then Y is an L- space. • (Question. Is �� (�′) torsion free for all rational homology spheres Y?) More L-space questions

• Conjecture. (Lidman, Moore) If Σ(�, �) is an L-space then � is an L- space. • Compatible with the L-space / non-left orderable conjecture. • Would be implied by a spectral sequence �� Σ �, � ⇒ �� (�). • Question. (Alishahi, Levine, Lidman, …) Does �� say anything about positive degree maps? • Maybe if there is a positive degree map � → �′ then rank �� ≥ rank �� ? • Question. What can you say about relationship between �� and � beyond L-spaces? The Poincaré conjecture

• Theorem. (Perelman, 2003) If � � = {1} then � ≅ � . • Conjecture. (Ozsváth-Szabó) If �� ≅ � then � is a connect sum of copies of the Poincaré homology sphere (and its mirror). • Together with L-space = not left orderable conjecture, this implies the Poincaré conjecture: • � � = {1} (trivially) implies � not left orderable. • By first conjecture, � not left orderable implies �� ≅ �. • By second conjecture, � is a connect sum of Poincaré spheres. • Thus, since � � = {1}, � = � . Thanks!

(Lots more to figure out…)